7. Let (V, ⟨⋅, ⋅⟩) be an inner product space and let R, S be subsets of V .(a) Prove that S ∩ S⊥ = 0

To prove that S ∩ S⊥ = {0}, we need to show that the only vector that is both in S and S⊥ is the zero vector.

Step 1: Assume that there is a vector v that is in both S and S⊥. This means that v is orthogonal to every vector in S, including itself.

Step 2: The definition of orthogonality in an inner product space is that the inner product of the two vectors is zero. So, ⟨v, v⟩ = 0.

Step 3: However, in an inner product space, the inner product of a vector with itself is equal to the square of its norm, ||v||^2. So, ||v||^2 = 0.

Step 4: The only vector with a norm of zero is the zero vector. Therefore, v must be the zero vector.

Step 5: So, the only vector that can be in both S and S⊥ is the zero vector, which means that S ∩ S⊥ = {0}. This completes the proof.